Which of course means the regular user isn't allowed to bind to their own
account and their password. So, I'm back to the drawing board. I still can't
figure out how to change slapd.conf to enable regular users to change their
own passwords...

Bugger...

It would help if you paid attention and reported *exactly* what you typed and
*exactly* what output you got from each command. For example, it is
*impossible* for this command:
ldappasswd -x -D cn=annem,dc=mydomain,dc=com -w newpassword
to return this result:
"additional info: unauthenticated bind (DN with no password) disallowed"